The index of dispersion is a statistic commonly used to detect departures from randomness of count data. Under the hypothesis of randomness, the true distribution of this statistic is unknown. The accuracy of large sample approximations is assessed by a Monte Carlo simulation. Further approximations by Pearson curves and infinite series expansions are investigated. Finally, the powers of the individual tests based on the likelihood ratio, the index of dispersion and Pearson's goodness-of-fit statistic are compared.